BSTJ 50: 4. April 1971: Maximum Power Transmission Between Two Reflector Antennas in the Fresnel Zone. (Chu, T.S.)

Copyright © 1971 American Telephone and Telegraph Company

The Bell System Technical Journal

Vol. SO, No. 4, April, 1971

Printed in U.S.A.

Maximum Power Transmission Between

Two Reflector Antennas in

the Fresnel Zone

By T. S. CHU

(Manuscript received August 11, 1970)

Power transfer between two ellipsoidal reflector antennas with com-
mon focal points and dual-mode feeds has been investigated. Asmming
a circularly symmetric feed pattern, the H-plane pattern of an open-
end circular waveguide excited by the TE n mode, the maximum trans-
mission- coefficient between reflector apertures is found to be within
0.5 percent of the value computed for transmission between two circular
apertures with optimum illumination desci'ibed by the generalized
prolate spheroidal function. Transmission loss between two reflector
antennas versus illumination taper is computed for various values of
the parameter \p = fka A a- 2 )/R]. The minimum transmission loss is
obtained as a compromise between feed spill-over and aperture trans-
mission efficiency.

In view of the inconvenience of building different ellipsoidal reflec-
tors for different antenna spacings in order to achieve maximum power
transfer, we examine the feasibility of using a defocused ellipsoid to
simulate ellipsoids of different focal lengths. The deviation of the
aperture phase distribution of a defocused ellipsoid from a required
spherical phase front is approximately given by an explicit expression.
A simple upper bound of the transmission loss due to small phase devi-
ation is obtained for a given maximum phase deviation.

I. INTRODUCTION

Millimeter waves have never been used in long-haul radio transmis-
sion systems because of rain attenuation. However, wideband transmis-
sion over short span (say less than 1 km) can be accomplished by
millimeter wave systems, for example, the Picturephone® distribution
in large cities proposed by R. Kompfner. The very high transmission

1407

1408 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1971

efficiency may leave a large margin for rain attenuation to insure high
reliability. Here the transmitting and receiving antennas may be
within the Fresnel zones of each other.

Maximum power transfer between two apertures in the Fresnel
region takes place when the field distribution appears as the lowest or-
der mode of a confocal open resonator. 1 - 2 Two ellipsoidal reflectors
with common focal points may provide the required aperture phase
distribution which is a spherical wave front with the center of curva-
ture located at the center of the other aperture. The optimum ampli-
tude distribution is a prolate spheroidal function for a rectangular
aperture or a generalized prolate spheroidal function for a circular
aperture. S. Takeshita 3 has shown that truncated gaussian illumination,
which is asymptotically identical to the generalized prolate spheroidal
function, may give a transmission efficiency between two circular aper-
tures almost as good as that using generalized prolate spheroidal illum-
ination. However, the truncated gaussian distribution only looks simpler
in terms of mathematical manipulation while its practical realization
appears not any easier than that of prolate speroidal functions. Fur-
thermore, if a lens or reflector is used to produce the spherical phase
front, the feed spill-over must be taken into account. The maximum
power transfer between the feeds of two reflector antennas will be ob-
tained as a compromise between feed spill-over loss and aperture trans-
mission efficiency. This procedure is similar to optimizing the gain of a
paraboloidal antenna. The aperture blocking effect can be made very
small by using the periscope type structure 4 which virtually eliminates
the feed supports. This paper will present the calculated results of
Fresnel zone transmission between two ellipsoidal reflector antennas
with dual-mode feeds. In view of the inconvenience of building differ-
ent ellipsoidal reflectors for different antenna spacings, we will also ex-
amine the feasibility of using a defocused ellipsoid to simulate ellip-
soids of different focal lengths.

II. MAXIMUM POWER TRANSFER

Neglecting the interaction between the antennas and assuming that
the tangential components of the electric and magnetic fields are re-
lated by the free space impedance at each point of the two apertures
A x and A 2 , the ratio of the received to transmitted power between two
apertures* at any separation can be shown 5 to be

* Not to be confused with transmission between two antennas which includes
spill-over of the feeds.

FRESNEL ZONE POWER TRANSFER

1409

Pr

1 /*:

Ja, Ja,

j E 2 ds t ds 2

(I)

\ 1 I I E[ | 2 dSy

E 2 | 2 ds. 4

where E[ and E 2 are tangential components of the aperture field dis-
tributions when A x and A 2 are transmitting respectively. Using the
small angle Fresnel approximation, the distance L may be approxi-
mated by

L = [R 2 + (x - tf + (y - t,) 2 ]*,

(2)

fl +

(x - £) 2 + (y - vY
2R

where the coordinate system is illustrated in Fig. 1. Then the near
field power transmission formula becomes

rp _ £-K -
Pt~

f A J A ^^[j H ^R yii) ] ^^^

R 2 \

iU E ^ ld 4L

/ , 'ds^j^ /■;, ,/

where

E, =SJ exp [-J 2 fi J '
E t -E t e X p\-]—^ J.

(3)

(4a)
(4b)

Fig. 1 — Two ellipsoidal reflectors with common focal points.

1410 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1971

If the aperture distributions are circularly symmetric, equation (3)
can be reduced to

( f E l (r)VrE 2 (r')Vr' Joiprr^Vpr/drdr'

Jo Jo -

I_ I «/Q ♦'O .

V |£ I E&) Vr | 2 *}{£ I E 2 (r') V? | 2 dr j

(5)

where p = (kata^/R for two circular apertures of radii a x and a* .
One notes that the transmitting and receiving apertures may have
different radii, while the optimum illumination function is identical
for the two apertures. When both Ej and E 2 are real, the Schwartz
principle gives the following condition for the maximization of equa-
tion (5)

j£ E(r) Vr=f E(r') Vr' J (prr') Vprr' dr' .

\p J

(6)

The subscript of E has been dropped because equation (6) is satisfied
by both Ex and E 2 . The above integral equation has been thoroughly
investigated by D. Slepian 6 and 0(r) = E(r)Vr is designated as a
generalized prolate spheroidal function. The optimum transmission
coefficient T taken from Slepian's work is given in Table I for reference.
Now a simple way of approximately realizing the optimum aperture
distribution appears to be the illumination of an ellipsoidal reflector by
a dual-mode feel. In the vicinity of the reflector, the reflected field will
follow the geometrical optics rays which are pointed toward the remote
focal point as shown in Fig. 1, and thus the required spherical phase
front will be created in the aperture. An experiment on dual-mode
apertures 7 , one to two wavelengths in diameter, showed the measured
patterns to be circularly symmetric and essentially in agreement with
the #-plane pattern of an open-end circular waveguide excited by
TEn mode, i.e.,

F(6) = \yji - riMY + cos 0J

J[(u sin 6) ,„,

_ ( u sin g \ a
\ 1.841/

where u is the circumference of the waveguide in wavelengths. Since
the ellipsoidal reflector is very closely a defocused paraboloid, as will
be shown in the next section, the space attenuation factor needed for
the variable distance from the feed to the reflector surface is essen-
tially the same as that of a paraboloid. Then the aperture distribution
is related to the feed pattern by

FRESNEL ZONE POWER TRANSFER

1411

E(r) = F(0) cos 2

(8)

where ar/2f = tan 6/2 and / is the focal length of the reflector. The
combination of equations (7) and (8) can be used to determine the
value of u for any corresponding illumination taper as plotted in Fig.
2. Substituting equation (8) into equation (5), numerical integration
will give the transmission coefficient between two ellipsoidal reflector
apertures excited by dual-mode feeds. The computed data have been
plotted in Fig. 3 for f/D = 0.5 and various values of the parameter p.
The maximum efficiency using dual-mode feed and excluding spill-over
has been tabulated in Table I for comparison with the optimum effi-
ciency using amplitude illumination of generalized prolate spheroidal
functions.

The agreement between the second and third columns in Table I is
indeed excellent. The insensitivity of the transmission coefficient to
small differences in illumination is not surprising in view of the sta-
tionary property of equation (5) . Table I does not show the maximum
efficiency with dual-mode feed for p = 2 and 10, because the mathe-
matical model in equation (7) for the dual-mode pattern has not been
experimentally verified for the circumference of the waveguide outside
the range 3 ^ it- ^ 6. However, this model covers the most interesting
range and demonstrates that efforts to synthesize a truncated gaussian
distribution are unwarranted. As far as maximum power transfer be-
tween two apertures is concerned, any aperture distribution which re-
sembles the generalized prolate spheroidal function may achieve prac-
tically the optimum transmission efficiency. As an example of tolerable
discrepancy, the dual-mode feed illumination function of a taper
strength which maximizes the transmission coefficient for p = 5 is com-
pared with the corresponding generalized prolate spheroidal function

Table I — Power Transfer Efficiency of Two
Circular Apertures

p = ka\a-i/R

Optimum Illumination
Efficiency

Dual Mode Illumination
Maximum Efficiency

2.0
3.0
4.0
5.0
10.0

0.630
0.887
0.975
0.995
1.000

0.886
0.972
0.992

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THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1971

in Fig. 4. One notes that the over-all similarity is sufficient to achieve
transmission coefficients differing by only 0.3 percent while the edge
illuminations differ by more than 3 dB.

Next we turn our attention to the spill-over loss of a feed. If the re-
flector is illuminated by a circularly symmetric feed pattern F(9),
then the fraction of the energy intercepted by the reflector will be

a = [' [F(d)] 2 sin 9 dd / f [F(0)] 2 sin 6 dd

(9)

where 6 is the half angle subtended by the reflector at the focus and
the back lobes have been neglected. Substituting equation (7) into
equation (9), we compute the total spill-over loss in decibels from
2 [10 logio (l/«)] as shown in Fig. 3. The presence of two feeds in the
system accounts for the factor two. It is seen that the spill-over loss is
as important as the power transfer loss between two reflector apertures
in determining the total transmission loss between two reflector an-
tennas. The minimum total transmission loss between feeds will always
be greater than the power transfer loss between the reflector apertures,
and always occurs at an illumination taper stronger than that for the
maximum transmission between two apertures. Any slight decrease in
transmission loss due to deviation of the aperture distribution from
the generalized prolate spheroidal function will certainly be swamped
by the feed spill-over loss. Figure 3 indicates that the illumination
taper corresponding to the maximum power transfer decreases as the
parameter p decreases. As the distance between two reflectors increases
toward the far zone condition p = 0, the optimum illumination taper

Fig. 2 — Relation between feed pattern parameter and taper.

FRESNEL ZONE POWER TRANSFER
2.5

2.0

1.5

1413

o

2.5

2.0 -

25 30 5 10

TAPER IN DECIBELS

Fig. 3 — Power transfer between ellipsoidal reflector antennas with dual mode
feeds, (a) ka^/R = 2, (b) ka^/R = 3, (c) ka^/R = 4, (d) hktb/R = 5.

will approach 12 dB which maximizes the Fraunhofer gain of a para-
boloidal antenna. 8 One notes that 12-dB aperture taper corresponds to
10-dB feed pattern taper, allowing a 2-dB space attenuation factor, for
0.5 f/D ratio. When f/D decreases to smaller values, the spill-over loss
curve will be shifted to the right because of increasing space attenua-
tion factor. Then the minimum of the sum will be also moved to the
right and upward. Large f/D ratio which requires narrow feed pattern
may cause significant loss due to aperture blocking which has been
neglected in the above calculations.

III. OPTIMUM DEFOCUSING OF AN ELLIPSOID

In the preceding section we have shown that ellipsoidal reflectors
are needed for maximum power transfer between two reflector antennas
in the Fresnel region. The required focal lengths of the ellipsoidal re-

1414

THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1971

1.0

0.8

0.4

0.2

ka, a ;

Qi0 {v)/Vr, <P 0i0 ir)\s a

GENERALIZED PROLATE
SPHEROIDAL FUNCTION

U= 5.25

19.2 dB TAPER

DUAL MODE FEED

0.4 0.6 0.8 1.0

T, NORMALIZED RADIUS

1.4

Fig, 4 — Comparison between optimum aperture distributions.

flectors are determined by the distance between the reflectors and the
desired f/D ratio. It is obviously inconvenient to build different ellip-
soidal reflectors for different antenna spacings ; however, approximat-
ing ellipsoids of different focal lengths by a defocused ellipsoid can be
an attractive possibility in some practical situations. The necessary
displacement of the feed along the axis will be first determined. Then
we will calculate the approximate phase deviation between the wave
front reflected from a defocused ellipsoid and the spherical wave front
in the aperture of a required ellipsoidal reflector. An upper bound of
the transmission loss will be obtained for a given maximum phase
deviation.

The equation of an ellipsoid can be written as

1 + R

z =

b-^ii

(10)

where / and E* are the two focal lengths. Now let us consider another
ellipsoid with corresponding focal lengths /' and R'

-^t-^RH-

(ii)

* R may be taken as any distance between /a — h and U , where /i and / 2 are
the two focal lengths of the ellipsoid. This slight ambiguity results from the
aperture approximation for reflector antenna, and is unimportant provided that
/a is orders of magnitude greater than /i .

FRESNEL ZONE POWER TRANSFER

1415

These two ellipsoids coincide at the tip P = 0. We will impose the con-
dition that the two ellipsoids also coincide at the edge p — a as shown
in Fig. 5. Taking the first three terms of the binomial expansion of
the square root in equations (10) and (11), we obtain an expression
for the required defocus distance :

= /'-/ =

f(R - R

2 [■ * (i)']

(12)

RR' 1/ ' \2fj
in which the approximations / <£ R and /' <SC R' have been used. Sub-

stituting equation (12) into equations (10) and
deviation between the two ellipsoids becomes

11), the approximate

1 a

AZ = ivi/r (1 - r)

a 2 (R - R')
RR'

(13)

where r = p/a is the normalized radius. Multiplying equation (13) by
the factor (1 4- cos 6) yields the phase deviation between the two wave
fronts reflected from the two ellipsoids as shown in Fig. 5. Since these
ellipsoids differ little from paraboloids, the relation tan 0/2 = ar/2f is
approximately valid. Then one arrives at the following expression

*r 2 (l - r 2 )

! + (!

a 2 R - R'
XR' R

(14)

The last factor in equation (14) is always less than unity when R
> R'. This factor approaches unity when R — » oo, i.e., approximating
an ellipsoid by a defocused paraboloid. The quantity inside the bracket
of equation (14) is the phase deviation 9 of a wave front produced by a

Fig. 5 — Geometry of defocused ellipsoid.

1416

THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1971

defocused paraboloid from a desired spherical wave front for a Fresnel
number (a 2 /\R) of unity, and has been plotted in Fig. 6 for various
values of f/D ratio. The maximum phase deviation is found to be lo-
cated at

When (a/2/) 2 «. 1, the above equation can be reduced to

2
P =

sh m

(15)

(16)

If the variation of antenna spacing covers a range from Ri to R 2 , the
optimum ellipsoid for minimizing the phase deviation of equation (14)
can be found by equating

-0.02 -

5 -0.04

<

> -0.06

-0.08

-0.10

= 0.75 ^//ll

_J).§ZS_^^ / 1 \

___o.5 ^^y^ / /

~~ \ ^s.

0.375^/ /

1 1 1

0.25/

1

0,2

0.4 0.6

r 7 NORMALIZED RADIUS

0.8

1.0

Fig. 6 — Aperture phase deviation between a defocused paraboloid and an
ellipsoid for unity Fresnel number.

FRESNEL ZONE POWER TRANSFER

1417

R 2 — R' _ R' — R\
R 2 R R t R

Solving the above equation, we have

2R 2 R 2

R' =

Ri t R 2

(17)

(18)

Substituting equation (18) into equation (14), the maximum phase
deviation becomes

\r\l - r 2 )

' - \%

a 2 R,
\R,

-R t

2R,

(19)

One notes that the above total phase deviation is twice the maximum
absolute magnitude of the phase error for the defocused ellipsoid.
In the Appendix an upper bound of the transmission loss due to small
phase error has been approximately determined as (w, + m 2 ) 2 for
optimum aperture distributions where m t and m 2 are the maximum
absolute magnitudes of phase error of the two apertures respectively. As
a numerical example, if the antenna spacing varies by a factor of 3,
equation (19) and Fig. 6 will give nu = m 2 = (%(8/\))-2ir = 0.03 for
a Fresnel number of unity and an f/D ratio of 0.5. Then the fractional
transmission loss due to phase error will be less than 0.4 percent.

IV. DISCUSSION

The above calculations have demonstrated the potential of optimum
Fresnel zone transmission between two antennas with ellipsoidal
reflectors illuminated by dual-mode feeds. In spite of the discrepancy
between the illumination function of a dual-mode feed and the
optimum illumination of a generalized prolate spheroidal function,
the maximum power transmission between reflector apertures is prac-
tically the same for the two cases. However, the feed spill-over loss is
as important as the transmission loss between reflector apertures in
determining the total transmission loss between reflector antennas.
More sophisticated feed design, such as the synthesis of more than
two modes, may reduce the spill-over loss. Here the resulting increase
in aperture blocking of the reflector is undesirable. The use of a lens
in place of a reflector would avoid aperture blocking but would give
rise to interface matching problems. Furthermore, the proper organiza-

1418 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1971

tion of many modes implies narrow bandwidth and stringent toler-
ances.

From a geometrical optics point of view, one is tempted to have the
ellipsoidal reflectors focused at the midpoint between them. This
scheme corresponds to a concentric resonator. The aperture distribu-
tion created by the two reflectors which are portions of the same
ellipsoid, as shown' in Fig. 1, is equivalent to a confocal resonator.
The diffraction loss of a concentric resonator is much greater than
than of a confocal resonator. The gaussian beam theory 10 predicts a
beam waist, i.e., an effective focal region, in the middle of the confocal
resonator. These observations indicate the failure of geometrical optics,
although the reflectors are in the Fresnel zones of each other. Con-
verting the reflected field into the aperture distribution employs
geometrical optics ray tracing only in the immediate vicinity of the
reflector. The validity of this procedure should be as good as that of
calculating the diffraction pattern from the aperture distribution of
a paraboloid al antenna.

The feasibility of using a defocused ellipsoid to simulate ellipsoids of
different focal lengths has been investigated. The optimum defocusing
of an ellipsoidal reflector for obtaining another spherical wavefront is
similar to that of defocusing a spherical reflector for obtaining an
approximate plane wave front. The explicit expression for phase
deviation shows its simple dependence on the f/D ratio, the Fresnel
number, and the variation of antenna spacing. Since a defocused
paraboloid is a special case of a defocused ellipsoid, the criteria ob-
tained here will also be useful for measuring the Fraunhofer radiation
pattern of a paraboloidal antenna in the Fresnel region. In particular,
it clarifies the inconsistency among various schemes for this latter
problem, proposed by D. K. Cheng. 9 ' 11

V. ACKNOWLEDGMENT

The author wishes to thank A. B. Crawford for suggesting this prob-
lem and for helpful discussions.

APPENDIX

An upper bound of the transmission loss due to small phase error
will be given below for nearly optimum aperture distributions. The
phase error may be either deterministic or random. A small phase
error simply introduces an additional factor exp(;A<£) into the

FRESNEL ZONE POWER TRANSFER

1419

numerator of equation (5), where A<£ = A(r) + A(r / ). For small
values of A<f>,

exp (jbf>) & 1 - i(A^>) 2 + jA<f>.
Then the modified equation (5) becomes

- = \ f f M l (r)M 2 ( r ')J (prr , )Vpr7 |l - ^ 1 drdr'
p \ Jo Jo \- & J

+ \ f f M 1 (r)M 2 (r')J (prr')Vpr7 A<t> dr dr'

(20)

where

M<(r) = -

EMV'r

[ \E,(r)Vr\ 2 dr

Jo

i = 1, 2.

(21)

(22)

If Ei and E 2 are optimum aperture distributions, then for any pertur-
bation factors A(r) and B(r') with maximum magnitudes m A and m B ,
the following inequality holds,

f f A(r)M 1 (r)B(r')M 2 (r , )J (prr')Vpr7 dr dr'

I •'0 J

^ m\m\ [ f M i (r)M 2 (r')Jo(prr') Vpr7 dr dr' , (23)

| Jo •'0

where A and B can be either of the following combinations
\A = A(r) \A = [A(r)] 2 U - 1

'? = A(r')' {B - 1 IB - [A(r')] 2

It follows that

7 1 ^ T nnt 1

opt T _ (», + m 2 ) 2 "| 2 ^

(24)

(25)

where mi = |A(r)| mnx and m^ = lAf/JIm^ . The maximum fractional
transmission loss due to small phase error is

1 - =r- ^ (w» + ™ 2 ) 2 1 -

i opt L

(m, + w 2 )

]

(26)

The above upper bound of the effect of phase error on the Fresnel
zone transmission loss represents a conservative estimate.

1420 THE BELL SYSTEM TECHNICAL JOURNAL, APRIL 1971

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■| CO

2 Heurtiey J C, "Maximum Power Transfer Between Finite Antennas," IEEE

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